A growth model based on the arithmetic Z-game

We present an evolutionary self-governing model based on the numerical atomic rule Z(a,b)=ab/gcd^(a,b)2, for a, b positive integers. Starting with a sequence of numbers, the initial generation _Gin, a new sequence is obtained by applying the Z-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix _TGin is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If _Gin is the sequence of positive integers, in the associated matrix remarkable are the distinguished geometrical figures called Z-solitons and the sinuous evolution of the size of numbers on the western edge. We observe that _TN∗ is close to the analogue free of Z-solitons matrix generated from an initial generation in which each natural number is replaced by its largest divisor that is a product of distinct primes. We describe the shape and the properties of this new matrix. N. J. A. Sloane raised a few interesting problems regarding the western edge of the matrix _TN∗. We solve one of them and present arguments for a precise conjecture on another.